Question

the equation of a parabola is 12y=(x-1)^2-48 identify the vertex , focus, and directrix of the parabola.

Answer 1

the equation of a vertically opening parabola is

(x - h)² = 4p(y - k)

where (h, k) are the coordinates of the vertex and p is the distance of the focus and directrix from the vertex. If 4p is positive parabola opens up and if 4p is negative parabola opens down

rearrange 12y = (x - 1)² - 48 into this form

(x - 1)² = 12y + 48

(x - 1)² = 12( y + 4) ← in standard form

Vertex = (1, - 4 )

4p = 12 ⇒ p = 3 ⇒ parabola opens upwards

the focus is above the vertex and the directrix is below the vertex in a vertically opening up parabola

Focus = (1, - 4 + 3 ) = (1, - 1)

Directrix has equation y = - 4 - 3 ⇒ y = - 7